Finite Group Databasebeta

← #13107114

Subgroups — #13107114

Order 1600.

1320 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 267 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1039C211yesmaximal, characteristic, complemented
2C2×#87902C211yesmaximal, characteristic, complemented
3C52×#120C211yesmaximal, characteristic, complemented
4#11717273C211yesmaximal, characteristic, complemented
5#11717259C212yesmaximal, complemented
6#11717255C211yesmaximal, characteristic, complemented
7D4×#1039C212yesmaximal, complemented
8#11717206C211yesmaximal, characteristic, complemented
9#11717201C212yesmaximal, complemented
10#11717198C212yesmaximal, complemented
11#11717197C211yesmaximal, characteristic, complemented
12C24×C52C2211yescharacteristic
13C2×C52×D4C2212yes—
14C2×C52×D4C2211yescharacteristic
15C2×C22⋊(C5⋊D5)C2212yes—
16C2×C22⋊(C5⋊D5)C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
18C22×#1039—24no—
19C22×#1039C2212yes—
20C22×#1039C2212yes—
21C22×#1039C2211yescharacteristic
22C22×#1039C2211yescharacteristic
23C22×#1039C2211yescharacteristic, complemented
24#87902—24no—
25C2×C4×C5⋊D5C2211yescharacteristic, complemented
26C2×#6045C2211yescharacteristic, complemented
27C52×C22⋊C4C2211yescharacteristic
28C52×C22⋊C4C2212yes—
29C22⋊#1039C2211yescharacteristic
30C22⋊#1039C2212yes—
31C22⋊#1039C2212yescomplemented
32#87808C2212yescomplemented
33C4⋊#1039C2212yescomplemented
34#87806C2212yescomplemented
35#87806C2212yescomplemented
36C4×#1039C2212yescomplemented
37#65920—56nomaximal
38C23×C52—22no—
39C23×C52—24no—
40C23×C52C2311yescharacteristic
41C23×C52C2311yescharacteristic
42C23×C52C2312yes—
43C22×C5⋊D5C2311yescharacteristic
44C52×D4—22no—
45C52×D4—24no—
46C2×C4×C52C2312yes—
47C2×C4×C52C2311yescharacteristic
48C22⋊(C5⋊D5)—22no—
49C22⋊(C5⋊D5)—24no—
50C22⋊(C5⋊D5)—22no—
51C2×#1039—22no—
52C2×#1039—22no—
53C2×#1039—24no—
54C2×#1039—24no—
55C2×#1039D414yescomplemented
56C2×#1039C2312yes—
57C2×#1039C2312yes—
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C4×C5⋊D5—22no—
62#6045—22no—
63C23×Dic5—56no—
64C2×#711—56no—
65C5×#120D516yescomplemented
66#4099—56no—
67#4085—512no—
68#4081—56no—
69D4×Dic5—512no—
70#4032—56no—
71#4027—512no—
72#4024—512no—
73#4023—56no—
74C22×C52—28no—
75C22×C52—22no—
76C22×C52—22no—
77C22×C52—22no—
78C22×C52—21no—
79C22×C52—22no—
80C22×C52C2×D412yes—
81C22×C52#4114yescomplemented
82C22×C52C2411yescharacteristic
83C2×C5⋊D5—21no—
84C2×C5⋊D5—22no—
85C4×C52—21no—
86C4×C52—22no—
87#1039—21no—
88#1039—22no—
89#1039—22no—
90#1039C2×D412yes—
91#1039C2×D412yes—
92C24×C5C2×D516yes—
93C2×C5×D4C2×D516yes—
94C2×C5×D4C2×D5112yes—
95C2×C22⋊D5—56no—
96C2×C22⋊D5—512no—
97C2×C22⋊D5—56no—
98C22×Dic5—56no—
99C22×Dic5—56no—
100C22×Dic5—512no—
101C22×Dic5—512no—
102C22×Dic5—56no—
103C22×Dic5—1024no—
104#711—1024no—
105C2×C4×D5—56no—
106C2×Dic10—56no—
107C5×C22⋊C4C2×D516yes—
108C5×C22⋊C4C2×D5112yes—
109C22⋊Dic5—56no—
110C22⋊Dic5—512no—
111C22⋊Dic5—512no—
112#685—512no—
113C4⋊Dic5—512no—
114#683—512no—
115#683—512no—
116C4×Dic5—512no—
117#525—251noSylow
118C2×C52—22no—
119C2×C52—24no—
120C2×C52—41no—
121C2×C52C22×D411yescharacteristic
122C2×C52C2×#4112yes—
123C5⋊D5—41no—
124C23×C5—212no—
125C23×C5—224no—
126C23×C5C22×D516yes—
127C23×C5C22×D516yes—
128C23×C5C22×D5112yes—
129C22×D5—56no—
130C5×D4—212no—
131C5×D4—224no—
132C2×C4×C5C22×D5112yes—
133C2×C4×C5C22×D516yes—
134C22⋊D5—1012no—
135C22⋊D5—1012no—
136C22⋊D5—1024no—
137C2×Dic5—524no—
138C2×Dic5—56no—
139C2×Dic5—56no—
140C2×Dic5—512no—
141C2×Dic5—512no—
142C2×Dic5—56no—
143C2×Dic5—1012no—
144C2×Dic5—1024no—
145C2×Dic5—1024no—
146C2×Dic5—1012no—
147C4×D5—1012no—
148Dic10—1012no—
149C2×#41—251no—
150C23×C4—251no—
151#123—251no—
152#123—251no—
153C22⋊Q8—252no—
154#121—252no—
155#120—251no—
156#120C5⋊D511yescharacteristic, complemented
157C4×D4—252no—
158C4×D4—252no—
159#117—251no—
160C52#52511yescharacteristic, complemented, Sylow
161C22×C5—248no—
162C22×C5—212no—
163C22×C5—212no—
164C22×C5—212no—
165C22×C5—26no—
166C22×C5—212no—
167C22×C5D4×D5112yes—
168C22×C5#711124yescomplemented
169C22×C5C23×D516yes—
170C2×D5—106no—
171C2×D5—1012no—
172C4×C5—26no—
173C4×C5—212no—
174Dic5—512no—
175Dic5—512no—
176Dic5—106no—
177Dic5—1012no—
178Dic5—1012no—
179C24C2×C5⋊D511yescharacteristic
180#41—504no—
181C2×Q8—251no—
182C2×D4—252no—
183C2×D4—251no—
184C2×D4—251no—
185C2×D4C2×C5⋊D512yes—
186C2×D4C2×C5⋊D511yescharacteristic
187C22×C4—251no—
188C22×C4—252no—
189C22×C4—251no—
190C22×C4—252no—
191C22×C4—251no—
192C22×C4—251no—
193C22×C4—504no—
194C4⋊C4—252no—
195C4⋊C4—252no—
196C4⋊C4—252no—
197C22⋊C4—251no—
198C22⋊C4—252no—
199C22⋊C4—252no—
200C22⋊C4—252no—
201C22⋊C4C2×C5⋊D511yescharacteristic
202C22⋊C4C2×C5⋊D512yes—
203C42—252no—
204C2×C5—212no—
205C2×C5—224no—
206C2×C5—46no—
207C2×C5C2×D4×D516yes—
208C2×C5C2×#711112yes—
209D5—206no—
210C23—22no—
211C23—24no—
212C23—251no—
213C23C22×C5⋊D511yescharacteristic
214C23C22×C5⋊D511yescharacteristic
215C23C22×C5⋊D512yes—
216Q8—502no—
217D4—22no—
218D4—24no—
219D4—502no—
220D4—502no—
221D4—504no—
222C2×C4—254no—
223C2×C4—251no—
224C2×C4—251no—
225C2×C4—252no—
226C2×C4—252no—
227C2×C4—251no—
228C2×C4—502no—
229C2×C4—504no—
230C2×C4—504no—
231C2×C4—502no—
232C2×C4—502no—
233C2×C4C22×C5⋊D512yes—
234C2×C4C22×C5⋊D511yescharacteristic
235C5#6592016yescomplemented
236C22—28no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—21no—
241C22—22no—
242C22—501no—
243C22—502no—
244C22D4×C5⋊D512yes—
245C22#8790214yescomplemented
246C22C23×C5⋊D511yescharacteristic
247C4—21no—
248C4—22no—
249C4—252no—
250C4—252no—
251C4—502no—
252C4—501no—
253C4—502no—
254C2—22no—
255C2—24no—
256C2—41no—
257C2—1001no—
258C2C2×D4×C5⋊D511yescharacteristic
259C2C2×#8790212yes—
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#120C2
hypercenter#120C5⋊D5
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #120

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×#1039C2semidirect
2. C2×#87902C2semidirect
3. C52×#120C2semidirect
4. #11717273C2semidirect
5. #11717259C2semidirect
6. #11717255C2semidirect
7. D4×#1039C2semidirect
8. #11717206C2semidirect
9. #11717201C2semidirect
10. #11717198C2semidirect
11. #11717197C2semidirect
17. C2×C22⋊(C5⋊D5)C22semidirect
23. C22×#1039C22semidirect
25. C2×C4×C5⋊D5C22semidirect
26. C2×#6045C22semidirect
31. C22⋊#1039C22semidirect
32. #87808C22semidirect
33. C4⋊#1039C22semidirect
34. #87806C22semidirect
35. #87806C22semidirect
36. C4×#1039C22semidirect
55. C2×#1039D4semidirect
65. C5×#120D5semidirect
81. C22×C52#41semidirect
156. #120C5⋊D5semidirect
160. C52#525semidirect
168. C22×C5#711semidirect
235. C5#65920semidirect
245. C22#87902semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×#10392
C2×#879022
C52×#1202
#117172732
#117172592
#117172552
D4×#10392
#117172062
#117172012
#117171982
#117171972
#659205